Arithmetic-Geometric Mean
المؤلف:
Abramowitz, M. and Stegu"The Process of the Arithmetic-Geometric Mean." §17.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dovern, I. A.
المصدر:
"The Process of the Arithmetic-Geometric Mean." §17.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
17-3-2019
3868
Arithmetic-Geometric Mean
The arithmetic-geometric mean 
of two numbers 
and 
(often also written 
or 
) is defined by starting with 
and 
, then iterating
until 
to the desired precision.
and 
converge towards each other since
But 
, so
 |
(5)
|
Now, add 
to each side
 |
(6)
|
so
 |
(7)
|
The top plots show 
for 
and 
for 
, while the bottom two plots show 
for complex values of 
.
The AGM is very useful in computing the values of complete elliptic integrals and can also be used for finding the inverse tangent.
It is implemented in the Wolfram Language as ArithmeticGeometricMean[a, b].
can be expressed in closed form in terms of the complete elliptic integral of the first kind 
as
 |
(8)
|
The definition of the arithmetic-geometric mean also holds in the complex plane, as illustrated above for 
.
The Legendre form of the arithmetic-geometric mean is given by
 |
(9)
|
where 
and
 |
(10)
|
Special values of 
are summarized in the following table. The special value
 |
(11)
|
(OEIS A014549) is called Gauss's constant. It has the closed form
where the above integral is the lemniscate function and the equality of the arithmetic-geometric mean to this integral was known to Gauss (Borwein and Bailey 2003, pp. 13-15).
 |
Sloane |
value |
 |
A068521 |
1.4567910310469068692... |
 |
A084895 |
1.8636167832448965424... |
 |
A084896 |
2.2430285802876025701... |
 |
A084897 |
2.6040081905309402887... |
The derivative of the AGM is given by
where 
, 
is a complete elliptic integral of the first kind, and 
is the complete elliptic integral of the second kind.
A series expansion for 
is given by
![agm(1,b)=-pi/(2ln(1/4b))+(pi[1+ln(1/4b)]b^2)/(8[ln(1/4b)]^2)+O(b^4).](http://mathworld.wolfram.com/images/equations/Arithmetic-GeometricMean/NumberedEquation8.gif) |
(16)
|
The AGM has the properties
Solutions to the differential equation
 |
(21)
|
are given by ![[agm(1+x,1-x)]^(-1)](http://mathworld.wolfram.com/images/equations/Arithmetic-GeometricMean/Inline69.gif)
and ![[agm(1,x)]^(-1)](http://mathworld.wolfram.com/images/equations/Arithmetic-GeometricMean/Inline70.gif)
.
A generalization of the arithmetic-geometric mean is
 |
(22)
|
which is related to solutions of the differential equation
 |
(23)
|
The case 
corresponds to the arithmetic-geometric mean via
The case 
gives the cubic relative
discussed by Borwein and Borwein (1990, 1991) and Borwein (1996). For 
, this function satisfies the functional equation
![I_3(a,b)=I_3((a+2b)/3,[b/3(a^2+ab+b^2)]^(1/3)).](http://mathworld.wolfram.com/images/equations/Arithmetic-GeometricMean/NumberedEquation12.gif) |
(28)
|
It therefore turns out that for iteration with 
and 
and
so
 |
(31)
|
where
 |
(32)
|
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "The Process of the Arithmetic-Geometric Mean." §17.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 571 and 598-599, 1972.
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.
Borwein, J. M. Problem 10281. "A Cubic Relative of the AGM." Amer. Math. Monthly 103, 181-183, 1996.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.
Borwein, J. M. and Borwein, P. B. "A Remarkable Cubic Iteration." In Computational Method & Function Theory: Proc. Conference Held in Valparaiso, Chile, March 13-18, 1989 (Ed. A. Dold, B. Eckmann, F. Takens, E. B. Saff, S. Ruscheweyh, L. C. Salinas, and R. S. Varga). New York: Springer-Verlag, 1990.
Borwein, J. M. and Borwein, P. B. "A Cubic Counterpart of Jacobi's Identity and the AGM." Trans. Amer. Math. Soc. 323, 691-701, 1991.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 906-907, 1992.
Sloane, N. J. A. Sequences A014549, A068521, A084895, A084896, and A084897 in "The On-Line Encyclopedia of Integer Sequences."
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